Satellite Formation-Keeping Using the Fundamental Equation in the Presence of Uncertainties in the System

Formation flying of satellites is considered as a key space technology because of its potential operational and/or financial benefits. In this paper a formation-keeping control scheme with attitude constraints is proposed in the presence of uncertainties in the masses and moments of inertia of the satellites. In formation-keeping, we assume that the satellites are to stay in their prescribed, desired orbits, and simultaneously, to point to a fixed target in space. In this study, we obtain the desired controller in a two-step process: we first obtain a controller for the nominal system, which is referred to the best assessment of the given real-life uncertain system. This controller can be analytically attained under the presumption that there are no uncertainties in the masses and moments of inertia of the satellites with the aid of a recent finding in analytical dynamics, called the fundamental equation. With this controller the system exactly follows the given constraint trajectories for the dynamical model assumed. Unlike previous studies, no approximations/linearizations are done related to the nonlinear nature of the system. However, this analytical result is correct only under the assumption that the modeling of the physical system has no errors and uncertainties. Consequently, we need to modify it to account for the uncertainties in our dynamical model and then the second step deals with the compensation of these model uncertainties. Using the constrained trajectory of the nominal system obtained in the first step as the tracking signal, a modified controller is developed, based on the concept of the sliding mode control. This modified controller guarantees satisfaction of the given orbital and attitude constraints when unknown, but bounded, mass and moment of inertia uncertainties are present. To demonstrate the effectiveness of the proposed control methodology, we consider a formation in which a leader satellite is in a circular orbit around the spherical Earth. A follower satellite is controlled in such a manner that it moves around the leader satellite in a projected circular orbit, simultaneously pointing towards the center of the Earth at all times. A numerical simulation demonstrates that the controller developed herein guarantees successful tracking of the given trajectory constraints even when the uncertainty is pertinent to the dynamical system.